Synchronizing an oscillatory medium: The speed of pacemaker-generated waves

Abstract

In an oscillatory medium, a region which oscillates faster than its surroundings can act as a source of outgoing waves. Such pacemaker-generated waves can synchronize the whole medium and are present in many physical and biological systems, where they are a means of transmitting information. Through numerical simulations, we quantify how the properties of the pacemaker and the underlying limit cycle determine the wave speed, as well as the speed with which they overtake the medium. We compare oscillators based on two of the main mechanisms that generate oscillations in biochemical systems: bistability and time delay. We show that these mechanisms produce oscillations whose wave propagation properties differ markedly. While both types of oscillatory media admit waves that propagate linearly outwards, the dependence of the wave speed on a timescale separation parameter is different between the two. If timescale separation is lost, waves no longer spread linearly. Finally, we quantify the effects of pacemaker size, the frequency difference with the surroundings, and the diffusion strength.

Analytical approximations for the speed of pacemaker-generated waves

“…1a). In our previous paper we showed that the envelope does not need to spread linearly [1]. This can happen when ε is large, or when the size of the pacemaker or its frequency difference with the outside medium is too small.…”

“…A numerical study of the influence of the diffusion constants can be found in Ref. [1]. To modify the characteristics of the underlying oscillator, we focus on a single parameter: ε, which determines the timescale separation.…”

“…In a previous paper [1], we described our numerical results on the speed of pacemaker-generated waves. There, we demonstrated the difference between an oscillator built on an underlying bistability and one in which the oscillations were generated by a time-delayed negative feedback loop.…”

“…In both systems, we investigate the influence of timescale separation on the resulting wave speed, as well as the effect of size and frequency of the pacemaker region. We compare our analytical estimates to numerical simulations which we described previously [1].…”

Mitotic waves in an import-diffusion model with multiple nuclei in a shared cytoplasm

“…It would be interesting to see how different types of oscillators (i.e. sinusoidal vs. relaxation oscillators) affect the wave properties, which has recently been characterized in the context of pacemaker-driven waves in reaction-diffusion systems [38]. The type of coupling used in the model offers another possibility to study in more detail.…”